/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 44 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   fac(0) -> 1
   fac(s(x)) -> *(s(x), fac(x))
   floop(0, y) -> y
   floop(s(x), y) -> floop(x, *(s(x), y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   1 -> s(0)
   fac(0) -> s(0)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
fac_1 > 1 > s_1 > 0
fac_1 > *_2 > +_2 > s_1 > 0
floop_2 > *_2 > +_2 > s_1 > 0


Status:
fac_1: multiset status
0: multiset status
1: multiset status
s_1: multiset status
*_2: multiset status
floop_2: [1,2]
+_2: multiset status

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   fac(0) -> 1
   fac(s(x)) -> *(s(x), fac(x))
   floop(0, y) -> y
   floop(s(x), y) -> floop(x, *(s(x), y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   1 -> s(0)
   fac(0) -> s(0)




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES